Question

In ΔАВC, ∠ABC = 90° and D is any point on BC. Prove that : AD² + BC² = AC² + BD².

Answer 1

Given:

In triangle ABC, ∠ABC = 90°.

D is any point on BC.

To Prove:

AD² + BC² = AC² + BD².

Proof [Step-wise]:

1. Since ∠ABC = 90° and D lies on BC, BA ⟂ BD. 

So, triangle ABD is right-angled at B.

By the Pythagorean theorem in ΔABD: 

AD² = AB² + BD²

2. Also, triangle ABC is right-angled at B.

By the Pythagorean theorem in ΔABC:

AC² = AB² + BC²

3. Subtract the second equation from the first:

AD² – AC²

= (AB² + BD²) – (AB² + BC²) 

= BD² – BC²

4. Rearranging gives

AD² + BC² = AC² + BD²